ECON2228 Notes 10 Christopher F Baum Boston College Economics - - PowerPoint PPT Presentation

ECON2228 Notes 10

Christopher F Baum

Boston College Economics

2014–2015

cfb (BC Econ) ECON2228 Notes 10 2014–2015 1 / 48

Serial correlation and heteroskedasticity in time series regressions

Chapter 12: Serial correlation and heteroskedasticity in time series regressions

What will happen if we violate the assumption that the errors are not serially correlated, or autocorrelated? We demonstrated that the OLS estimators are unbiased, even in the presence of autocorrelated errors, as long as the explanatory variables are strictly exogenous. This is analogous to our results in the case of heteroskedasticity, where the presence of heteroskedasticity alone does not cause bias nor inconsistency in the OLS point estimates. However, following that parallel argument, we will be concerned with the properties of our interval estimates and hypothesis tests in the presence of autocorrelation.

cfb (BC Econ) ECON2228 Notes 10 2014–2015 2 / 48

Serial correlation and heteroskedasticity in time series regressions

OLS is no longer BLUE in the presence of serial correlation, and the OLS standard errors and test statistics are no longer valid, even

• asymptotically. Consider a first-order Markov error process:

ut = ρut−1 + et, |ρ| < 1 (1) where the et are uncorrelated random variables with mean zero and constant variance.

cfb (BC Econ) ECON2228 Notes 10 2014–2015 3 / 48

Serial correlation and heteroskedasticity in time series regressions

What will be the variance of the OLS slope estimator in a simple y on x regression model? For simplicity let us center the x series so that ¯ x = 0. Then the OLS estimator will be: b1 = β1 + T

t=1 xtut

SSTx where SSTx is the sum of squares of the x series.

cfb (BC Econ) ECON2228 Notes 10 2014–2015 4 / 48

Serial correlation and heteroskedasticity in time series regressions

In computing the variance of b1, conditional on x, we must account for the serial correlation in the u process: Var (b1) = 1 SST 2

x

Var T

• t=1

xtut

• =

1 SST 2

x

T

i=1 x2 t Var(ut)+

2 T−1

t=1

T−1

j=1 xtxt−jE

• utut−j
• =

σ2 SSTx + 2

• σ2

SST 2

x

T−1

• t=1

T−1

• j=1

ρjxtxt−j where σ2 = Var(ut) and we have used the fact that E

• utut−j
• = Cov
• utut−j
• = ρjσ2 in the derivation.

cfb (BC Econ) ECON2228 Notes 10 2014–2015 5 / 48

Serial correlation and heteroskedasticity in time series regressions

Notice that the first term in this expression is merely the OLS variance

• f b1 in the absence of serial correlation. When will the second term

be nonzero? When ρ is nonzero, and the x process itself is autocorrelated, this double summation will have a nonzero value. As nothing prevents the explanatory variables from exhibiting autocorrelation (and in fact many explanatory variables take on similar values through time) the only way in which this second term will vanish is if ρ is zero, and u is not serially correlated. In the presence of serial correlation, the second term will cause the standard OLS variances of

• ur regression parameters to be biased and inconsistent.

cfb (BC Econ) ECON2228 Notes 10 2014–2015 6 / 48

Serial correlation and heteroskedasticity in time series regressions

In most applications, when serial correlation arises, ρ is positive, so that successive errors are positively correlated. In that case, the second term will be positive as well. Recall that this expression is the true variance of the regression parameter; OLS will only consider the first term. In that case OLS will seriously underestimate the variance of the parameter, and the t−statistic will be much too high. If on the other hand ρ is negative, so that successive errors result from an “overshooting” process, then we may not be able to determine the sign of the second term, since odd terms will be negative and even terms will be positive. Surely, though, it will not be zero. Thus the consequence of serial correlation in the errors, particularly if the autocorrelation is positive, will render the standard t− and F−statistics useless.

cfb (BC Econ) ECON2228 Notes 10 2014–2015 7 / 48

Serial correlation in the presence of lagged dependent variables

Serial correlation in the presence of lagged dependent variables

A case of particular interest, even in the context of simple y on x regression, is that where the explanatory variable is a lagged dependent variable. Suppose that the conditional expectation of yt is linear in its past value: E (yt|yt−1) = β0 + β1yt−1. We can always add an error term to this relation, and write it as yt = β0 + β1yt−1 + ut (2)

cfb (BC Econ) ECON2228 Notes 10 2014–2015 8 / 48

Serial correlation in the presence of lagged dependent variables

Let us first assume that the error is “well behaved,” i.e. E (ut|yt−1) = 0, so that there is no correlation between the current error and the lagged value of the dependent variable. In this setup the explanatory variable cannot be strictly exogenous, since there is a contemporaneous correlation between yt and ut by construction. In evaluating the consistency of OLS in this context we are concerned with the correlation between the error and yt−1, not the correlation with yt, yt−2, and so on. In this case, OLS would still yield unbiased and consistent point estimates, with biased standard errors, as we derived above, even if the u process was serially correlated.

cfb (BC Econ) ECON2228 Notes 10 2014–2015 9 / 48

Serial correlation in the presence of lagged dependent variables

But it is often claimed that the joint presence of a lagged dependent variable and autocorrelated errors, OLS will be inconsistent. This arises, as it happens, from the assumption that the u process in (2) follows a particular autoregressive process, such as the first-order Markov process in (1). If this is the case, then we do have a problem of inconsistency, but it is arising from a different source: the misspecification of the dynamics of the model.

cfb (BC Econ) ECON2228 Notes 10 2014–2015 10 / 48

Serial correlation in the presence of lagged dependent variables

If we combine (2) with (1), we really have an AR(2) model for yt, since we can lag (2) one period and substitute it into (1) to rewrite the model as: yt = β0 + β1yt−1 + ρ (yt−1 − β0 − β1yt−2) + et = β0 (1 − ρ) + (β1 + ρ) yt−1 − ρβ1yt−2 + et = α0 + α1yt−1 + α2yt−2 + et (3) so that the conditional expectation of yt properly depends on two lags

• f y, not merely one. Thus the estimation of (2) via OLS is indeed

inconsistent, but the reason for that inconsistency is that y is correctly modelled as AR(2).

cfb (BC Econ) ECON2228 Notes 10 2014–2015 11 / 48

Serial correlation in the presence of lagged dependent variables

The AR(1) model is seen to be a dynamic misspecification of (3). As is always the case, the omission of relevant explanatory variables will cause bias and inconsistency in OLS estimates, especially if the excluded variables are correlated with the included variables. In this case, that correlation will almost surely be meaningful. To arrive at consistent point estimates of this model, we merely need add yt−2 to the estimated equation. That does not deal with the inconsistent interval estimates, which will require a different strategy.

cfb (BC Econ) ECON2228 Notes 10 2014–2015 12 / 48

Testing for first-order serial correlation

Testing for first-order serial correlation

As the presence of serial correlation invalidates our standard hypothesis tests and interval estimates, we should be concerned about testing for it. First let us consider testing for serial correlation in the k−variable regression model with strictly exogenous regressors, which rules out, among other things, lagged dependent variables.

cfb (BC Econ) ECON2228 Notes 10 2014–2015 13 / 48

Testing for first-order serial correlation

The simplest structure which we might posit for serially correlated errors is AR(1), the first order Markov process, as given in (1). Let us assume that et is uncorrelated with the entire past history of the u process, and that et is homoskedastic. The null hypothesis is H0 : ρ = 0 in the context of (1).

cfb (BC Econ) ECON2228 Notes 10 2014–2015 14 / 48

Testing for first-order serial correlation

If we could observe the u process, we could test this hypothesis by estimating (1) directly. Under the maintained assumptions, we can replace the unobservable ut with the OLS residual vt. Thus a regression of the OLS residuals on their own lagged values, vt = κ + ρvt−1 + ǫt, t = 2, ...T (4) will yield a t− test. That regression can be run with or without an intercept, and the robust option may be used to guard against violations of the homoskedasticity assumption. It is only an asymptotic test, though, and may not have much power in small samples.

cfb (BC Econ) ECON2228 Notes 10 2014–2015 15 / 48

Testing for first-order serial correlation

A very common strategy in considering the possibility of AR(1) errors is the Durbin–Watson test, which is also based on the OLS residuals: DW = T

t=2 (vt − vt−1)2

T

t=1 v2 t

(5)

cfb (BC Econ) ECON2228 Notes 10 2014–2015 16 / 48

Testing for first-order serial correlation

Simple algebra shows that the DW statistic is closely linked to the estimate of ρ from the large-sample test: DW ≃ 2 (1 − ˆ ρ) (6) ˆ ρ ≃ 1 − DW 2 The relationship is not exact because of the difference between (T − 1) terms in the numerator and T terms in the denominator of the DW test.

cfb (BC Econ) ECON2228 Notes 10 2014–2015 17 / 48

Testing for first-order serial correlation

The difficulty with the DW test is that the critical values must be evaluated from a table, since they depend on both the number of regressors (k) and the sample size (n), and are not unique: for a given level of confidence, the table contains two values, dL and dU. If the computed value falls below dL, the null is clearly rejected. If it falls above dU, there is no cause for rejection. But in the intervening region, the test is inconclusive. The test cannot be used on a model without a constant term, and it is not appropriate if there are any lagged dependent variables.

cfb (BC Econ) ECON2228 Notes 10 2014–2015 18 / 48

Testing for first-order serial correlation

In the presence of one or more lagged dependent variables, an alternative statistic may be used: Durbin’s h statistic, which merely amounts to augmenting (4) with the explanatory variables from the

• riginal regression. This test statistic may readily be calculated in

Stata with the estat durbinalt command.

cfb (BC Econ) ECON2228 Notes 10 2014–2015 19 / 48

Testing for higher-order serial correlation

Testing for higher-order serial correlation

One of the disadvantages of tests for AR(1) errors is that they consider precisely that alternative hypothesis. In many cases, if there is serial correlation in the error structure, it may manifest itself in a more complex relationship, involving higher-order autocorrelations; e.g. AR(p). A logical extension to the test described in (4) and the Durbin “h” test is the Breusch–Godfrey test, which considers the null of nonautocorrelated errors against an alternative that they are AR(p).

cfb (BC Econ) ECON2228 Notes 10 2014–2015 20 / 48

Testing for higher-order serial correlation Breusch–Godfrey and Q tests

This test can readily be performed by regressing the OLS residuals on p lagged values, as well as the regressors from the original model. The test is the joint null hypothesis that those p coefficients are all zero, which can be considered as another T × R2 Lagrange multiplier (LM) statistic, analogous to White’s test for heteroskedasticity. The test may easily be performed in Stata using the estat bgodfrey command. You must specify the lag order p to indicate the degree of autocorrelation to be considered. If p = 1, the test is essentially Durbin’s “h” statistic.

cfb (BC Econ) ECON2228 Notes 10 2014–2015 21 / 48

Testing for higher-order serial correlation Breusch–Godfrey and Q tests

An even more general test often employed on time series regression models is the Box–Pierce or Ljung–Box Q statistic, or “portmanteau test,” which has the null hypothesis that the error process is “white noise,” or nonautocorrelated, versus the alternative that it is not well behaved. The “Q” test evaluates the autocorrelation function of the errors, and in that sense is closely related to the Breusch–Godfrey test. That test evaluates the conditional autocorrelations of the residual series, whereas the “Q” statistic uses the unconditional autocorrelations.

cfb (BC Econ) ECON2228 Notes 10 2014–2015 22 / 48

Testing for higher-order serial correlation Breusch–Godfrey and Q tests

The “Q” test can be applied to any time series as a test for “white noise,” or randomness. For that reason, it is available in Stata as the command wntestq. This test is often reported in empirical papers as an indication that the regression models presented therein are reasonably specified. Any of these tests may be used to evaluate the hypothesis that the errors exhibit serial correlation, or nonindependence. But caution should be exercised when their null hypotheses are rejected. It is very straightforward to demonstrate that serial correlation may be induced by simple misspecification of the equation. For instance, if you model a relationship as linear when it is curvilinear, or when it represents exponential growth, the linear form is misspecified.

cfb (BC Econ) ECON2228 Notes 10 2014–2015 23 / 48

Testing for higher-order serial correlation Breusch–Godfrey and Q tests

Many time series models are misspecified in terms of inadequate dynamics: that is, the relationship between y and the regressors may involve many lags of the regressors. If those lags are mistakenly

• mitted, the equation suffers from misspecification bias, and the

regression residuals will reflect the missing terms. In this context, a visual inspection of the residuals is often useful. User-written Stata routines such as tsgraph, sparl and particularly

• frtplot should be employed to better understand the dynamics of

the regression function. Each may be located and installed with Stata’s ssc command, and each is well documented with on-line help.

cfb (BC Econ) ECON2228 Notes 10 2014–2015 24 / 48

Testing for higher-order serial correlation Breusch–Godfrey and Q tests

. summarize rs r20 Variable Obs Mean

• Std. Dev.

Min Max rs 526 7.651513 3.553109 1.561667 16.18 r20 526 8.863726 3.224372 3.35 17.18 . eststo, ti("OLS VCE"):regress D.rs LD.r20, vsquish Source SS df MS Number of obs = 524 F( 1, 522) = 52.88 Model 13.8769739 1 13.8769739 Prob > F = 0.0000 Residual 136.988471 522 .262430021 R-squared = 0.0920 Adj R-squared = 0.0902 Total 150.865445 523 .288461654 Root MSE = .51228 D.rs Coef.

• Std. Err.

t P>|t| [95% Conf. Interval] r20 LD. .4882883 .0671484 7.27 0.000 .356374 .6202027 _cons .0040183 .022384 0.18 0.858

• .0399555

.0479921 (est1 stored)

cfb (BC Econ) ECON2228 Notes 10 2014–2015 25 / 48

Testing for higher-order serial correlation Breusch–Godfrey and Q tests

Breusch–Godfrey and Q tests

. predict double eps, residual (2 missing values generated) . estat bgodfrey, lags(6) Breusch-Godfrey LM test for autocorrelation lags(p) chi2 df Prob > chi2 6 17.237 6 0.0084 H0: no serial correlation . wntestq eps Portmanteau test for white noise Portmanteau (Q) statistic = 82.3882 Prob > chi2(40) = 0.0001

cfb (BC Econ) ECON2228 Notes 10 2014–2015 26 / 48

Correcting for serial correlation with strictly exogenous regressors

Correcting for serial correlation with strictly exogenous regressors

As OLS cannot provide consistent interval estimates in the presence of autocorrelated errors, how should we proceed? If we have strictly exogenous regressors (in particular, no lagged dependent variables), we may be able to obtain an appropriate estimator through transformation of the model.

cfb (BC Econ) ECON2228 Notes 10 2014–2015 27 / 48

Correcting for serial correlation with strictly exogenous regressors

If the errors follow the AR(1) process in (1), we determine that Var(ut) = σ2

e/

• 1 − ρ2

. Consider a simple y on x regression with autocorrelated errors following an AR(1) process. Then simple algebra will show that the quasi-differenced equation (yt − ρyt−1) = (1 − ρ) β0 + β1 (xt − ρxt−1) + (ut − ρut−1) (7) will have nonautocorrelated errors, as the error term in this equation is in fact et, by assumption well behaved.

cfb (BC Econ) ECON2228 Notes 10 2014–2015 28 / 48

Correcting for serial correlation with strictly exogenous regressors

This quasi-differencing transformation can only be applied to

• bservations 2, ..., T, but we can write down the first observation in

static terms to complete that, plugging in a zero value for the time-zero value of u. This extends to any number of explanatory variables, as long as they are strictly exogenous; we just quasi-difference each, and use the quasi-differenced version in an OLS regression.

cfb (BC Econ) ECON2228 Notes 10 2014–2015 29 / 48

Correcting for serial correlation with strictly exogenous regressors

How can we employ this strategy when we do not know the value of ρ? It turns out that the feasible generalized least squares (GLS) estimator

• f this model merely replaces ρ with a consistent estimate, ˆ

ρ. The resulting model is asymptotically appropriate, even if it lacks small sample properties. We can derive an estimate of ρ from OLS residuals, or from the calculated value of the Durbin–Watson statistic on those residuals. Most commonly, if this technique is employed, we use an algorithm that implements an iterative scheme, revising the estimate of ρ in a number of steps to derive the final results.

cfb (BC Econ) ECON2228 Notes 10 2014–2015 30 / 48

Correcting for serial correlation with strictly exogenous regressors

One common methodology is the Prais–Winsten estimator, which makes use of the first observation, transforming it separately. It may be used in Stata via the prais command. That same command may also be used to employ the Cochrane–Orcutt estimator, a similar iterative technique that ignores the first observation. In a large sample, it will not matter if one observation is lost. This estimator can be executed using the corc option of the prais command. We do not expect these estimators to provide the same point estimates as OLS, as they are working with a fundamentally different model. If they provide similar point estimates, the FGLS estimator is to be preferred, as its standard errors are consistent. However, in the presence of lagged dependent variables, more complicated estimation techniques are required.

cfb (BC Econ) ECON2228 Notes 10 2014–2015 31 / 48

Correcting for serial correlation with strictly exogenous regressors

An aside on first differencing. An alternative to employing the feasible GLS estimator, in which a value of ρ inside the unit circle is estimated and used to transform the data, would be to first difference the data: that is, transform the left and right hand side variables into differences. This would indeed be the proper procedure to follow if it was suspected that the variables possessed a unit root in their time series representation. If the value of ρ in (1) is strictly less than 1 in absolute value, first differencing approximates that value, since differencing is equivalent to imposing ρ = 1 on the error process. If the process’s ρ is quite different from 1, first differencing is not as good a solution as applying the FGLS estimator.

cfb (BC Econ) ECON2228 Notes 10 2014–2015 32 / 48

Correcting for serial correlation with strictly exogenous regressors

Also note that if you difference a standard regression equation in y, x1, x2... you derive an equation that does not have a constant term. A constant term in an equation in differences corresponds to a linear trend in the levels equation. Unless the levels equation already contains a linear trend, applying differences to that equation should result in a model without a constant term.

cfb (BC Econ) ECON2228 Notes 10 2014–2015 33 / 48

Correcting for serial correlation with strictly exogenous regressors

. eststo, ti("GLS VCE"): prais D.rs LD.r20, nolog vsquish Prais-Winsten AR(1) regression -- iterated estimates Source SS df MS Number of obs = 524 F( 1, 522) = 25.73 Model 6.56420242 1 6.56420242 Prob > F = 0.0000 Residual 133.146932 522 .25507075 R-squared = 0.0470 Adj R-squared = 0.0452 Total 139.711134 523 .2671341 Root MSE = .50505 D.rs Coef.

• Std. Err.

t P>|t| [95% Conf. Interval] r20 LD. .3495857 .068912 5.07 0.000 .2142067 .4849647 _cons .0049985 .0272145 0.18 0.854

• .0484649

.0584619 rho .1895324 Durbin-Watson statistic (original) 1.702273 Durbin-Watson statistic (transformed) 2.007414 (est2 stored)

cfb (BC Econ) ECON2228 Notes 10 2014–2015 34 / 48

Robust inference in the presence of autocorrelation

Robust inference in the presence of autocorrelation

Just as we utilized the “White” heteroskedasticity-consistent standard errors to deal with heteroskedasticity of unknown form, we may generate estimates of the standard errors that are robust to both heteroskedasticity and autocorrelation. Why would we want to do this rather than explicitly take account of the autocorrelated errors via the feasible generalized least squares estimator described earlier? If we doubt that the explanatory variables may be considered strictly exogenous, then the FGLS estimates will not even be consistent, let alone efficient.

cfb (BC Econ) ECON2228 Notes 10 2014–2015 35 / 48

Robust inference in the presence of autocorrelation Newey–West standard errrors

Also, FGLS is usually implemented in the context of an AR(1) model, since it is much more complex to apply it to a more complex AR

• structure. But higher-order autocorrelation in the errors may be quite
• plausible. Robust methods may take account of that behavior.

The methodology to compute what are often termed heteroskedasticity- and autocorrelation-consistent (HAC) standard errors was developed by Newey and West; thus they are often referred to as Newey–West standard errors.

cfb (BC Econ) ECON2228 Notes 10 2014–2015 36 / 48

Robust inference in the presence of autocorrelation Newey–West standard errrors

Unlike the White standard errors, which require no judgment to calculate, the Newey–West standard errors must be calculated conditional on a choice of maximum lag. They are calculated from a distributed lag of the OLS residuals, and one must specify the longest lag at which autocovariances are to be computed. Normally a lag length exceeding the periodicity of the data will suffice; e.g. at least 4 for quarterly data, 12 for monthly data, etc. The Newey–West (HAC) standard errors may be readily calculated for any OLS regression using Stata’s newey command. You must provide the “option” lag( ), which specifies the maximum lag order, and your data must be tsset (that is, known to Stata as time series data).

cfb (BC Econ) ECON2228 Notes 10 2014–2015 37 / 48

Robust inference in the presence of autocorrelation Newey–West standard errrors

As the Newey-West formula involves an expression in the squares of the residuals which is identical to White’s formula (as well as a second term in the cross-products of the residuals), these robust estimates subsume White’s correction. Newey-West standard errors in a time series context are robust to both arbitrary autocorrelation (up to the

• rder of the chosen lag) as well as arbitrary heteroskedasticity.

cfb (BC Econ) ECON2228 Notes 10 2014–2015 38 / 48

Robust inference in the presence of autocorrelation Newey–West standard errrors

Computation of Newey–West standard errors

. eststo, ti("Newey-West"): newey D.rs LD.r20, lag(6) vsquish Regression with Newey-West standard errors Number of obs = 524 maximum lag: 6 F( 1, 522) = 35.74 Prob > F = 0.0000 Newey-West D.rs Coef.

• Std. Err.

t P>|t| [95% Conf. Interval] r20 LD. .4882883 .0816725 5.98 0.000 .3278412 .6487354 _cons .0040183 .0256542 0.16 0.876

• .0463799

.0544166 (est3 stored)

cfb (BC Econ) ECON2228 Notes 10 2014–2015 39 / 48

Robust inference in the presence of autocorrelation Newey–West standard errrors

Comparison of OLS, GLS, Newey–West estimates

. esttab, nonum mti se star(* 0.1 ** 0.05 *** 0.01) OLS VCE GLS VCE Newey-West LD.r20 0.488*** 0.350*** 0.488*** (0.0671) (0.0689) (0.0817) _cons 0.00402 0.00500 0.00402 (0.0224) (0.0272) (0.0257) N 524 524 524 Standard errors in parentheses * p<0.1, ** p<0.05, *** p<0.01

cfb (BC Econ) ECON2228 Notes 10 2014–2015 40 / 48

Heteroskedasticity in the time series context

Heteroskedasticity in the time series context

Heteroskedasticity can also occur in time series regression models; its presence, while not causing bias nor inconsistency in the point estimates, has the usual effect of invalidating the standard errors, t−statistics, and F−statistics, just as in the cross-sectional case. As the Newey–West standard error formula subsumes the White (robust) standard error component, if the Newey–West standard errors are computed, they will also be robust to arbitrary departures from

• homoskedasticity. However, the standard tests for heteroskedasticity

assume independence of the errors, so if the errors are serially correlated, those tests will not generally be correct. It thus makes sense to test for serial correlation first (using a heteroskedasticity–robust test if it is suspected), correct for serial correlation, and then apply a test for heteroskedasticity.

cfb (BC Econ) ECON2228 Notes 10 2014–2015 41 / 48

Heteroskedasticity in the time series context The ARCH model

In the time series context, it may be quite plausible that if heteroskedasticity—variations in volatility in a time series process—exists, it may itself follow an autoregressive pattern. This can be termed a dynamic form of heteroskedasticity, in which Engle’s ARCH (autoregressive conditional heteroskedasticity) model applies. The simplest ARCH model, the ARCH(1), may be written as: yt = β0 + β1zt + ut E

• u2

t |ut−1, ut−2, ...

• =

E

• u2

t |ut−1

• = α0 + α1u2

t−1

cfb (BC Econ) ECON2228 Notes 10 2014–2015 42 / 48

Heteroskedasticity in the time series context The ARCH model

The second line is the conditional variance of ut given that series’ past history, assuming that the u process is serially uncorrelated. As conditional variances must be positive, this only makes sense if α0 > 0 and α1 ≥ 0. We can rewrite the second line as: u2

t = α0 + α1u2 t−1 + υt

which then appears as an autoregressive model in the squared errors, with stability condition α1 < 1. When α1 > 0, the squared errors contain positive serial correlation, even though the errors themselves do not.

cfb (BC Econ) ECON2228 Notes 10 2014–2015 43 / 48

Heteroskedasticity in the time series context The ARCH model

If this sort of process is evident in the regression errors, what are the consequences? First of all, OLS are still BLUE. There are no assumptions on the conditional variance of the error process that would invalidate the use of OLS in this context. But we may want to explicitly model the conditional variance of the error process, since in many financial series the movements of volatility are of key importance (for instance, option pricing via the standard Black–Scholes formula requires an estimate of the volatility of the underlying asset’s returns, which may well be time–varying).

cfb (BC Econ) ECON2228 Notes 10 2014–2015 44 / 48

Heteroskedasticity in the time series context The ARCH model

Estimation of ARCH models, of which there are many flavors, with the most common extension being Bollerslev’s GARCH (generalized ARCH), may be performed via Stata’s arch command. Tests for ARCH, which are based on the squared residuals from an OLS regression, are provided by Stata’s estat archlm command.

cfb (BC Econ) ECON2228 Notes 10 2014–2015 45 / 48

Heteroskedasticity in the time series context The ARCH model

Test for ARCH effects

. regress D.rs LD.r20, vsquish Source SS df MS Number of obs = 524 F( 1, 522) = 52.88 Model 13.8769739 1 13.8769739 Prob > F = 0.0000 Residual 136.988471 522 .262430021 R-squared = 0.0920 Adj R-squared = 0.0902 Total 150.865445 523 .288461654 Root MSE = .51228 D.rs Coef.

• Std. Err.

t P>|t| [95% Conf. Interval] r20 LD. .4882883 .0671484 7.27 0.000 .356374 .6202027 _cons .0040183 .022384 0.18 0.858

• .0399555

.0479921 . estat archlm, lag(6) LM test for autoregressive conditional heteroskedasticity (ARCH) lags(p) chi2 df Prob > chi2 6 13.361 6 0.0377 H0: no ARCH effects vs. H1: ARCH(p) disturbance

cfb (BC Econ) ECON2228 Notes 10 2014–2015 46 / 48

Heteroskedasticity in the time series context The ARCH model

Estimation of ARCH(1)

. arch D.rs LD.r20, vsquish nolog arch(1) ARCH family regression Sample: 1952m5 - 1995m12 Number of obs = 524 Distribution: Gaussian Wald chi2(1) = 50.57 Log likelihood = -370.6064 Prob > chi2 = 0.0000 OPG D.rs Coef.

• Std. Err.

z P>|z| [95% Conf. Interval] rs r20 LD. .4458543 .0626973 7.11 0.000 .3229699 .5687387 _cons

• .0081822

.0235846

• 0.35

0.729

• .0544071

.0380427 ARCH arch L1. .3888359 .0729199 5.33 0.000 .2459155 .5317562 _cons .1819778 .0085672 21.24 0.000 .1651864 .1987692

cfb (BC Econ) ECON2228 Notes 10 2014–2015 47 / 48

Heteroskedasticity in the time series context The ARCH model

Estimation of GARCH(1,1)

. arch D.rs LD.r20, vsquish nolog arch(1) garch(1) ARCH family regression Sample: 1952m5 - 1995m12 Number of obs = 524 Distribution: Gaussian Wald chi2(1) = 54.58 Log likelihood = -368.9344 Prob > chi2 = 0.0000 OPG D.rs Coef.

• Std. Err.

z P>|z| [95% Conf. Interval] rs r20 LD. .4524499 .0612444 7.39 0.000 .332413 .5724867 _cons

• .0169603

.0224823

• 0.75

0.451

• .0610247

.0271041 ARCH arch L1. .3843838 .0727441 5.28 0.000 .241808 .5269595 garch L1.

• .0770956

.0200969

• 3.84

0.000

• .1164847
• .0377064

_cons .2037547 .0120402 16.92 0.000 .1801563 .227353

cfb (BC Econ) ECON2228 Notes 10 2014–2015 48 / 48